Schaum's Discrete Mathematics by Seymour Lipschutz

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Then a; b and b; c belong to S for every S. Since each S is transitive, a; c belongs to S for every S. Hence, a; c P T and T is transitive. 16. Let R be a relation on a set A, and let P be a property of relations, such as, symmetry and transitivity. Then P will be called R-closable if P satis®es the following two conditions: (1) There is a P-relation S containing R. (2) The intersection of P-relations is a P-relation. (a) Show that symmetry and transitivity are R-closable for any relation R.

4: RÃ is the transitive closure of a relation R. Suppose A is a ®nite set with n elements. Then we show in Chapter 8 on directed graphs that RÃ  R  R2  Á Á Á  Rn This gives us the following result. 5: Let R be a relation on a set A with n elements. Then transitive R  R  R2  Á Á Á  Rn Finding transitive R can take a lot of time when A has a large number of elements. An ecient way for doing this will be described in Chapter 8. Here we give a simple example where A has only three elements.

R3 is not symmetric since 1; 3 P R3 but 3; 1 P = R3 . The R1 is not symmetric since 1; 2 P R1 but 2; 1 P other relations are symmetric. 7 are symmetric. The relation c is symmetric since if line a is perpendicular to line b then b is perpendicular to a. Also, k is symmetric since if line a is parallel to line b then b is parallel to a. The other relations are not symmetric. For example, 3 4 but 4M3; f1; 2g f1; 2; 3g but f1; 2; 3g \ f1; 2g, and 2j6 but 6j=2. A relation R on a set A is antisymmetric if whenever aRb and bRa then a  b, that is, if whenever a; b, b; a P R then a  b.